### Ultracold atoms and

### neutron-rich matter in nuclei and astrophysics

### Achim Schwenk

**NORDITA program “Pushing the boundaries with cold atoms” **

Stockholm, Jan. 23, 2013

### Outline

Advances in nuclear forces

3N forces and neutron-rich nuclei

with J.D. Holt, J. Menendez, T. Otsuka, T. Suzuki 3N forces and neutron matter/stars

with K. Hebeler, T. Krüger, I. Tews, J.M. Lattimer, C.J. Pethick

Neutron polaron and density functionals with M. Forbes, A. Gezerlis,

K. Hebeler, T. Lesinski

### Large scattering lengths: Universal properties at low densities

strongly-interacting dilute

1/k_{F}
a_{s }
R

strong interactions via Feshbach resonances large for neutrons dilute Fermi system with large scattering length has universal properties

only Fermi momentum or density sets scale

physics is independent of interaction/system details:

from dilute neutron matter to resonant ^{6}Li or ^{40}K atoms in traps

Greiner et al. (2003)

### Chiral Effective Field Theory for nuclear forces

NN 3N 4N

Weinberg, van Kolck, Kaplan, Savage, Wise, Epelbaum, Kaiser, Machleidt, Meissner,…

Separation of scales: low momenta breakdown scale ~500 MeV

limited resolution at low energies,
can expand in powers (Q/Λ_{b})^{n}

LO, n=0 - leading order,

NLO, n=2 - next-to-leading order,…

expansion parameter ~ 1/3

(compare to multipole expansion for a charge distribution)

### Chiral Effective Field Theory for nuclear forces

NN 3N 4N

Separation of scales: low momenta breakdown scale ~500 MeV

include long-range pion physics few short-range couplings,

fit to experiment once

systematic: can work to desired

accuracy and obtain error estimates

Weinberg, van Kolck, Kaplan, Savage, Wise, Epelbaum, Kaiser, Machleidt, Meissner,…

### Chiral Effective Field Theory for nuclear forces

NN 3N 4N

Separation of scales: low momenta breakdown scale ~500 MeV

6Li fermions 2 spin states

from M. Zwierlein

large scattering length physics

neutrons with same density, temperature and spin polarization have the same properties!

### Chiral Effective Field Theory for nuclear forces

NN 3N 4N

Separation of scales: low momenta breakdown scale ~500 MeV

Ni et al., Nature (2010)

pion tensor/dipole interactions + …

→ compare to cold polar molecules

### Chiral Effective Field Theory for nuclear forces

NN 3N 4N

Separation of scales: low momenta breakdown scale ~500 MeV

include long-range pion physics few short-range couplings,

fit to experiment once

systematic: can work to desired

accuracy and obtain error estimates

Weinberg, van Kolck, Kaplan, Savage, Wise, Epelbaum, Kaiser, Machleidt, Meissner,…

### Why are there three-body forces?

tidal effects lead to 3-body forces in earth-sun-moon system

### Why are there 3N forces?

Nucleons are finite-mass composite particles, can be excited to resonances

dominant contribution from Δ(1232 MeV)

+ many shorter-range parts

chiral effective field theory (EFT)

Delta-less (Δ is treated as heavy): + shorter-range parts
**EFT provides a systematic and powerful approach for 3N forces **

tidal effects lead to 3-body forces in earth-sun-moon system

### Chiral Effective Field Theory and many-body forces

NN 3N 4N

Separation of scales: low momenta breakdown scale ~500 MeV

consistent NN-3N interactions

3N,4N: only 2 new couplings to N^{3}LO

c_{i} from πN and NN Meissner et al. (2007)

single-Δ: c_{1}=0, c_{3}=-c_{4}/2=-3 GeV^{-1}

c_{D}, c_{E} fit to ^{3}H, ^{4}He properties only

Weinberg, van Kolck, Kaplan, Savage, Wise, Epelbaum, Kaiser, Machleidt, Meissner,…

### Chiral Effective Field Theory for nuclear forces

NN 3N 4N

Separation of scales: low momenta breakdown scale ~500 MeV

c_{D}, c_{E} don’t contribute for neutrons
because of Pauli principle and

pion coupling to spin, also for c_{4}

Hebeler, AS (2010)

**all 3- and 4-neutron forces are **
**predicted to N**^{3}**LO! **

Weinberg, van Kolck, Kaplan, Savage, Wise, Epelbaum, Kaiser, Machleidt, Meissner,…

O F

### The oxygen anomaly

without 3N forces, NN interactions too attractive

O F

### The oxygen anomaly - not reproduced without 3N forces

many-body theory based on two-nucleon forces:

drip-line incorrect at ^{28}O

fit to experiment

28O

16O ^{24}O

### The shell model - impact of 3N forces

include ‘normal-ordered’ 2-body part of 3N forces (enhanced by core A) leads to repulsive interactions between valence neutrons

contributions from residual three valence-nucleon
interactions suppressed by E_{ex}/E_{F} ~ N_{valence}/N_{core}

Friman, AS (2011)

### Oxygen isotopes - impact of 3N forces

include ‘normal-ordered’ 2-body part of 3N forces (enhanced by core A) leads to repulsive interactions between valence neutrons

contributions from residual three valence-nucleon
interactions suppressed by E_{ex}/E_{F} ~ N_{valence}/N_{core}

Friman, AS (2011)

d_{3/2} orbital remains unbound from ^{16}O to ^{28}O

microscopic explanation of the oxygen anomaly Otsuka et al. (2010)

### Three-body forces and magic numbers

no N=28 magic number from microscopic NN forces

Zuker, Poves,…

Holt et al. (2010), Holt, Menendez, AS, in prep.

52Ca is 1.75 MeV more bound

compared to atomic mass evaluation

Gallant et al. (2012)

behavior of two-neutron separation
energy S_{2n} and odd-even staggering Δ_{n}
agrees with NN+3N predictions

### new

^{51,52}

### Ca TITAN measurements

28 29 30 31 32

Neutron Number N 0

1 2 3

∆ n

(3) (MeV)

28 29 30 31 32

8 10 12 14 16 18

S 2n (MeV)

AME2003 TITAN

NN+3N (MBPT) NN+3N (emp)

TITAN+

AME2003

### Neutron matter and neutron stars

**empirical**

### Impact of 3N forces on nuclear matter

chiral 3N forces fit to light nuclei predict nuclear matter saturation with theoretical uncertainties

Hebeler et al. (2011), Bogner et al. (2005)

**empirical**

### Impact of 3N forces on neutron matter

neutron matter is simpler system,
only long-range parts of 3N forces
contribute (c_{1} and c_{3})

Hebeler, AS (2010)

scales as in universal regime at low densities, cold atoms provide anchor point

neutron matter

### Chiral Effective Field Theory and many-body forces

NN 3N 4N

Separation of scales: low momenta breakdown scale ~500 MeV

consistent NN-3N interactions

3N,4N: only 2 new couplings to N^{3}LO

c_{i} from πN and NN Meissner et al. (2007)

single-Δ: c_{1}=0, c_{3}=-c_{4}/2=-3 GeV^{-1}

c_{D}, c_{E} fit to ^{3}H, ^{4}He properties only

Weinberg, van Kolck, Kaplan, Savage, Wise, Epelbaum, Kaiser, Machleidt, Meissner,…

### Impact of 3N forces on neutron matter

neutron matter uncertainties

dominated by 3N forces (c_{3} coupling)

Hebeler, AS (2010)

3N

### Impact of 3N forces on neutron matter

neutron matter uncertainties

dominated by 3N forces (c_{3} coupling)

Hebeler, AS (2010)

other microscopic calculations within band (but without uncertainties)

3N

cold atoms/QMC

Gezerlis, Carlson (2009)

### Impact of 3N forces on neutron matter

neutron matter uncertainties

dominated by 3N forces (c_{3} coupling)

Hebeler, AS (2010)

Problem: many equations of state

not consistent with neutron matter results

3N

cold atoms/QMC

Gezerlis, Carlson (2009)

### Symmetry energy and pressure of neutron matter

neutron matter band predicts
symmetry energy S_{v} and

its density dependence L comparison to experimental and observational constraints

Lattimer, Lim (2012)

neutron matter constraints

H: Hebeler et al. (2010) and in prep.

G: Gandolfi et al. (2011)

predicts correlation

but not range of S_{v} and L

### Chiral Effective Field Theory for nuclear forces

NN 3N 4N

Separation of scales: low momenta breakdown scale ~500 MeV

c_{D}, c_{E} don’t contribute for neutrons
because of Pauli principle and

pion coupling to spin, also for c_{4}

Hebeler, AS (2010)

**all 3- and 4-neutron forces are **
**predicted to N**^{3}**LO! **

Weinberg, van Kolck, Kaplan, Savage, Wise, Epelbaum, Kaiser, Machleidt, Meissner,…

### Complete N

^{3}

### LO calculation of neutron matter

Tews, Krüger, Hebeler, AS (2013).

### Complete N

^{3}

### LO calculation of neutron matter

first complete N^{3}LO result

includes uncertainties from bare NN, 3N, 4N

Tews, Krüger, Hebeler, AS (2013).

direct measurement of neutron star mass from increase in signal travel time near companion J1614-2230

most edge-on binary
pulsar known (89.17°)
+ massive white dwarf
companion (0.5 M_{sun})
heaviest neutron star
with 1.97±0.04 M_{sun }

Nature (2010)

### Discovery of the heaviest neutron star

Equation of state/pressure for neutron-star matter (includes small Y_{e,p})

pressure below nuclear densities agrees with standard crust equation of state only after 3N forces are included

### Impact on neutron stars

Hebeler et al. (2010) and in prep.Equation of state/pressure for neutron-star matter (includes small Y_{e,p})

pressure below nuclear densities agrees with standard crust equation of state only after 3N forces are included

extend uncertainty band to higher densities using piecewise polytropes allow for soft regions

### Impact on neutron stars

Hebeler et al. (2010) and in prep.13.0 13.5 14.0

log_{10} [g / cm^{3}]
31

32 33 34 35 36 37

log10P [dyne/cm2 ]

1

2

3

with c_{i} uncertainties

crust

crust EOS (BPS) neutron star matter

12 23

1

### Pressure of neutron star matter

constrain polytropes by causality and require to support 1.97 M_{sun} star

low-density pressure sets scale, chiral EFT interactions provide strong constraints, ruling out many model equations of state

14.2 14.4 14.6 14.8 15.0 15.2 15.4

log_{10} [g / cm^{3}]
33

34 35 36

log10P [dyne/cm2 ]

WFF1 WFF2 WFF3 AP4 AP3 MS1 MS3 GM3 ENG PAL GS1 GS2

14.2 14.4 14.6 14.8 15.0 15.2 15.4

33 34 35 36

PCL2 SQM1 SQM2 SQM3 PS

### Pressure of neutron star matter

constrain polytropes by causality and require to support 1.97 M_{sun} star

low-density pressure sets scale, chiral EFT interactions provide strong
constraints, ruling out many model equations of state_{ }

**central densities for 1.4 M**_{sun}** star: 1.7-4.4 ρ**_{0}** **

14.2 14.4 14.6 14.8 15.0 15.2 15.4

log_{10} [g / cm^{3}]
33

34 35 36

log10P [dyne/cm2 ]

WFF1 WFF2 WFF3 AP4 AP3 MS1 MS3 GM3 ENG PAL GS1 GS2

14.2 14.4 14.6 14.8 15.0 15.2 15.4

33 34 35 36

PCL2 SQM1 SQM2 SQM3 PS

### Pressure of neutron star matter

constrain polytropes by causality and require to support 1.97 M_{sun} star

low-density pressure sets scale, chiral EFT interactions provide strong constraints, ruling out many model equations of state

darker blue band for 2.4 M_{sun} star

14.2 14.4 14.6 14.8 15.0 15.2 15.4

log_{10} [g / cm^{3}]
33

34 35 36

log10P [dyne/cm2 ]

WFF1 WFF2 WFF3 AP4 AP3 MS1 MS3 GM3 ENG PAL GS1 GS2

14.2 14.4 14.6 14.8 15.0 15.2 15.4

33 34 35 36

PCL2 SQM1 SQM2 SQM3 PS

### Neutron star radius constraints

uncertainty from many-body forces and general extrapolation

constrains neutron star radius: 9.9-13.8 km for M=1.4 M_{sun }(±15% !)
consistent with extraction from X-ray burst sources Steiner et al. (2010)

provides important constraints for EOS for core-collapse supernovae

8 10 12 14 16

Radius [km]

0 0.5 1 1.5 2 2.5 3

Mass [M sun]

causality

### Neutron-star merger and gravitational waves

explore sensitivity to neutron-rich matter in neutron-star merger and gw signal

Bauswein, Janka (2012) and A. Bauswein et al., arXiv:1204.1888.

### Neutron polaron

calculated with QMC and effective field theory methods, polaron energy increases due to effective range

used in SN sim.

Forbes, Gezerlis, Hebeler, Lesinski, AS, in prep.

### Neutron polaron and density functionals

Neutron polaron provides constraints for nuclear density functional, most state-of-the-art functionals underpredict polaron energy

used in SN sim.

Forbes, Gezerlis, Hebeler, Lesinski, AS, in prep.

### Summary

Chiral effective field theory interactions provide strong constraints for neutron-rich nuclei/matter, 3N forces are a frontier

key to explain why ^{24}O is the heaviest oxygen isotope

key for neutron-rich nuclei: Ca isotopes and magic numbers

3N forces are dominant uncertainty of neutron (star) matter below nuclear densities, constrains neutron-star radii and equation of state neutron polaron constrains nuclear density functional

cold atoms provide anchor points at low densities